Local well-posedness of the 1D degenerate hyperbolic-Poisson coupled PKS system

This paper investigates the well-posedness of local smooth solutions for the one-dimensional free boundary problem of the Patlak–Keller–Segel (PKS) model. By introducing a mixed velocity, the PKS model can be written into a form of the hyperbolic-Poisson coupled system, which becomes a degenerate system on the free boundary. We introduce the Lagrange transformation and convert the free boundary problem into an initial boundary value problem. Then, the degeneracy of this system can be described by the function [Formula: see text] of the initial density [Formula: see text] under Lagrangian coordinates, which is equivalent to the distance function near the boundary points. By utilizing the Hardy inequality and weighted Sobolev spaces, we establish high-order a priori estimates of local smooth solutions for any slow diffusion component [Formula: see text] overcoming the degeneracy issues at the free boundary. This confirms the existence and uniqueness of the local smooth solutions.